Symplectic rational blow-ups on rational 4-manifolds

TL;DR

This paper proves that symplectic rational blow-ups preserve rationality in 4-manifolds and extends classical algebraic geometry results to symplectic degenerations with quotient singularities.

Contribution

It establishes that symplectic rational blow-ups of rational 4-manifolds remain rational, and generalizes algebraic geometry degeneration results to symplectic settings.

Findings

Rational blow-up preserves rationality in symplectic 4-manifolds.

Degenerations with quotient singularities are rational surfaces.

Extension of classical algebraic results to symplectic geometry.

Abstract

We prove that if a symplectic 4-manifold $X$ becomes a rational 4-manifold after applying rational blow-down surgery, then the symplectic 4-manifold $X$ is originally rational. That is, a symplectic rational blow-up of a rational symplectic $4$-manifold is again rational. As an application we show that a degeneration of a family of smooth rational complex surfaces is a rational surface if the degeneration has at most quotient surface singularities, which generalizes slightly a classical result of [B\u{a}descu, 1986] in algebraic geometry under a mild additional condition.